# Geometry, MAT 631, Fall 2009

Elementary geometry from an advanced standpoint.

This course is part of the Mathematics Teacher Transformation Institute, an NSF Math & Science Partnership program.

 Geometry, MAT631, Fall 2009 Grading Policy: 11 in class projects, no exams. Homework: read over material, and finish projects. Prerequisites: participation in MTTI. Professor C. Sormani Office: Gillet Hall 200B Email: sormanic (at) member.ams.org Webpage: http://comet.lehman.cuny.edu/sormani

Course Description: Classical Euclidean Geometry using postulates and axioms will be combined with a more modern transformation perspective in this course tailored to provide teachers with an in depth understanding of the geometry in New York State's new high school mathematics curriculum.

Text: (to be provided by MTTI)

College Geometry: A discovery approach, by David C. Kay, 2nd Edition.
Appendix B has a review of high school geometry.
Appendix C discusses Geometers Sketchpad (we won't cover).
Appendix F has a list of axioms for quick reference (bring to class).
This book approaches geometry using the three classical geometries: Euclidean, spherical and hyperbolic. In this course we focus on the Euclidean part as this directly relates to the NYS curriculum, particular the Geometry Regents Exam which includes proofs.
Other Supplies: (not provided, bring your own)
Grades: The grade will be based on the projects. Projects must be completed with two column proofs and neat hand drawn sketches using straight edges and compasses. Proofs must be done in two columns: Statements and Justifications. The projects will be graded harshly with deductions taken for errors. However, the lowest grades will be dropped and lots of extra credit will be assigned. Projects will be done in class and submitted the following week at the beginning of class. Most projects will include at least one very difficult problems.

Syllabus: The projects in the following syllabus may be changed or reordered as we would like the projects to be relevant to the participants. For the proofs we will be following Kay's set of axioms, which correspond more closely to the NYS curriculum and modern mathematics than Euclid's ancient set of axioms.

Sept 9: Initial Meeting:
Required Incoming Assessment (necessary for statistical evaluation).
Extra Credit: Early History of pi (16-17) and Archimedes' Calculation (p19-21)
Sept 16: First Axioms
2.3-2.4 Points, Lines, Planes, Distances, (read pp 70-73, pp77-82)
Circles, Rays, Angles, Right Angles, Perpendicular Lines
Project 1: Write axioms and theorems with proofs assigned in class in order and drawings of each axiom and theorem.
Sept 23: Measuring Lengths and Angles and Convex Sets:
2.4-2.6 Ruler Postulate, Protractor Postulate, Convex Sets and Crossbar Theorem (read pp 83-85, 90-99, 103-110)
Project 2: Read pp 83-85 & 90-99 then prove (1) Angle Construction Theorem and (2) Unique Perpendicular Line Theorem. Read pp 103-104 then prove (3) Intersection of Convex sets is Convex but not unions. Read pp105-106 and prove (4)Thm 1. Read p 107 and prove (5) Pasch's Thm. Extra Credit: Read pp 108-109 and prove (6) Thm 3 and (7) Crossbar Thm. All proofs must be written in two columns. Some proofs may be found in the book or on the web. You must understand what you have written and be able to present it.
Sept 30: Congruent Triangles:
3.1, 3.3-3.4: SAS, ASA, etc
Project 3: Handout in class. Otherwise do the following:
(1) Read handout and fill in justifications for the proof of SSS. Read 123-124 and 139-142 on SAS Postulate/Hypothesis and ASA Theorem, Ex 1-3, Isosceles Triangle defn and Theorem, Construction of the perpendicular bisector.
(2) On graph paper using compass and straight edge, construct the perpendicular bisector of segment AB, where A=(1,3) and B=(5,7), then where A=(-2,4) and B=(-6, 2).
(3) Suppose A,B, and C are noncolinear, prove there exists a unique line perpendicular to line BC passing through A. Hint: Find a point D such that BD=AD and measure angle CBA= measure and angle CBD.
(4) Angle bisector construction: Given angle ABC with AB=BC, show that if a point X has AX=CX, then ray BX is the angle bisector. You may use Theorem SSS to do this proof.
(5) Write up proof of SSA as stated on page 175 (or in handout). Read proof of ASA in handout.
Note we pospone the proof of AAS which can be proven quickly once we know the sum of the angles of a triangle is 180 degrees. That cannot be proven until we introduce Euclid's Parallel Postulate. Right now everything we've proven is true for more general geometries like the sphere and hyperbolic space. For a proof of AAS in these more general spaces, you may read 3.6 if you wish.
Extra Credit: Resubmit Pythagoras proofs adding in angle justifications.
Oct 7: Concurrent Lines Oct 14: Parallel Postulate:
Parallel Lines, Angle Sum Theorem, Parallelograms 4.1-4.2
Project 5: Parallel Postulate Project pages 1-3
Extra Credit: Solid Geometry: Read 7.1, write out two column proofs of Theorems 1-3, Relate to the NYS Standards Concepts G.G.1-G.G.9,
Oct 21: Parallelograms:
Parallelograms, rhombuses and squares
Lesson begins with examples of bridges, why are some rigid while others collapse and why equilateral triangles make straight lines.
Project 6: Parallel Postulate Project page 4
Oct 28: Similar triangles:
Similar triangles 4.3, proportions, SOHCAHTOA,
Project 7: Similar Triangles Project (a version with diagrams in on my door).
Nov 4: Pythagoras and its Converse: Nov 11: Symmetries and Isometries: Nov 18: Circles, Arcs and Chords:
Circles 3.8 and 4.5, NYS Standards GG49-GG53.
Project 10: Circles Project pages 1-3. Page 4 is Extra Credit.
Nov 25: Coordinate Geometry and Vectors:
4.3, 4.7 Coordinate Plane, Quadrants, Lines through the Origin, Perpendicular Lines
Project 11: Coordinate Plane Project
Dec 2: Transformations, Areas :
Shift Isometry and Lines, Skews/Shears and Dilations, and Area 4.6, Skews and Dilations 5.1, 5.4
Project 12: Transformations and Area. Complete both projects as project 12.
Dec 9: Solid Geometry
7.1-7.4 p 297
Project 13: Intuitive Solid Geometry Project
Dec 16: Discussion of Direct Application in Classroom:
Bring NYS Standards and Questions